MDPI - Publisher of Open Access Journals

15 pages, 307 KiB

Open AccessArticle

Fuzzy Treatment for Meromorphic Classes of Admissible Functions Connected to Hurwitz–Lerch Zeta Function

by Ekram E. Ali, Rabha M. El-Ashwah, Abeer M. Albalahi and Rabab Sidaoui

Axioms 2025, 14(7), 523; https://doi.org/10.3390/axioms14070523 - 8 Jul 2025

Viewed by 288

Fuzzy differential subordinations, a notion taken from fuzzy set theory and used in complex analysis, are the subject of this paper. In this work, we provide an operator and examine the characteristics of meromorphic functions in the punctured open unit disk that are [...] Read more.

Fuzzy differential subordinations, a notion taken from fuzzy set theory and used in complex analysis, are the subject of this paper. In this work, we provide an operator and examine the characteristics of meromorphic functions in the punctured open unit disk that are related to a class of complex parameter operators. Complex analysis ideas from geometric function theory are used to derive fuzzy differential subordination conclusions. Due to the compositional structure of the operator, some pertinent classes of admissible functions are studied through the application of fuzzy differential subordination. Full article

(This article belongs to the Special Issue New Developments in Geometric Function Theory, 3rd Edition)

15 pages, 298 KiB

Open AccessFeature PaperArticle

The Approximation of Analytic Functions Using Shifts of the Lerch Zeta-Function in Short Intervals

by Antanas Laurinčikas

Axioms 2025, 14(6), 472; https://doi.org/10.3390/axioms14060472 - 17 Jun 2025

Viewed by 318

Abstract

In this paper, we obtain approximation theorems of classes of analytic functions by shifts

L (λ, α, s + i τ)

of the Lerch zeta-function for

τ \in [T, T + H]

where [...] Read more.

In this paper, we obtain approximation theorems of classes of analytic functions by shifts

L (λ, α, s + i τ)

of the Lerch zeta-function for

τ \in [T, T + H]

where

H \in [T^{27 / 82}, T^{1 / 2}]

. The cases of all parameters,

λ, α \in (0, 1]

, are considered. If the set

{\log (m + α) : m \in N_{0}}

is linearly independent over

Q

, then every analytic function in the strip

{s = σ + i t \in C : σ \in (1 / 2, 1)}

is approximated by the above shifts. Full article

23 pages, 340 KiB

Open AccessArticle

Third-Order Fuzzy Subordination and Superordination on Analytic Functions on Punctured Unit Disk

by Ekram E. Ali, Georgia Irina Oros, Rabha M. El-Ashwah and Abeer M. Albalahi

Axioms 2025, 14(5), 378; https://doi.org/10.3390/axioms14050378 - 17 May 2025

Viewed by 324

Abstract

This work’s theorems and corollaries present new third-order fuzzy differential subordination and superordination results developed by using a novel convolution linear operator involving the Gaussian hypergeometric function and a previously studied operator. The paper reveals methods for finding the best dominant and best [...] Read more.

This work’s theorems and corollaries present new third-order fuzzy differential subordination and superordination results developed by using a novel convolution linear operator involving the Gaussian hypergeometric function and a previously studied operator. The paper reveals methods for finding the best dominant and best subordinant for the third-order fuzzy differential subordinations and superordinations, respectively. The investigation concludes with the assertion of sandwich-type theorems connecting the conclusions of the studies conducted using the particular methods of the theories of the third-order fuzzy differential subordination and superordination, respectively. Full article

(This article belongs to the Special Issue Advances in Geometric Function Theory and Related Topics)

12 pages, 257 KiB

Open AccessArticle

Partial Sums of the Hurwitz and Allied Functions and Their Special Values

by Nianliang Wang, Ruiyang Li and Takako Kuzumaki

Mathematics 2025, 13(9), 1469; https://doi.org/10.3390/math13091469 - 29 Apr 2025

Viewed by 323

Abstract

We supplement the formulas for partial sums of the Hurwitz zeta-function and its derivatives, producing more integral representations and generic definitions of important constants. Then, these are used, coupled with the functional equation for the completed zeta-function to clarify the results of Choudhury, [...] Read more.

We supplement the formulas for partial sums of the Hurwitz zeta-function and its derivatives, producing more integral representations and generic definitions of important constants. Then, these are used, coupled with the functional equation for the completed zeta-function to clarify the results of Choudhury, giving rise to closed expressions for the Riemann zeta-function and its derivatives. Full article

(This article belongs to the Special Issue Analytic Methods in Number Theory and Allied Fields)

16 pages, 312 KiB

Open AccessArticle

Joint Approximation by the Riemann and Hurwitz Zeta-Functions in Short Intervals

by Antanas Laurinčikas

Symmetry 2024, 16(12), 1707; https://doi.org/10.3390/sym16121707 - 23 Dec 2024

Cited by 1 | Viewed by 795

Abstract

In this study, the approximation of a pair of analytic functions defined on the strip

{s = σ + i t \in C : 1 / 2 < σ < 1}

by shifts [...] Read more.

In this study, the approximation of a pair of analytic functions defined on the strip

{s = σ + i t \in C : 1 / 2 < σ < 1}

by shifts

(ζ (s + i τ), ζ (s + i τ, α))

,

τ \in R

, of the Riemann and Hurwitz zeta-functions with transcendental

α

in the interval

[T, T + H]

with

T^{27 / 82} ⩽ H ⩽ T^{1 / 2}

was considered. It was proven that the set of such shifts has a positive density. The main result was an extension of the Mishou theorem proved for the interval

[0, T]

, and the first theorem on the joint mixed universality in short intervals. For proof, the probability approach was applied. Full article

(This article belongs to the Section Mathematics)

14 pages, 293 KiB

Open AccessArticle

Fuzzy Subordination Results for Meromorphic Functions Associated with Hurwitz–Lerch Zeta Function

by Ekram E. Ali, Georgia Irina Oros, Rabha M. El-Ashwah, Abeer M. Albalahi and Marwa Ennaceur

Mathematics 2024, 12(23), 3721; https://doi.org/10.3390/math12233721 - 27 Nov 2024

Cited by 2 | Viewed by 881

Abstract

The notion of the fuzzy set was incorporated into geometric function theory in recent years, leading to the emergence of fuzzy differential subordination theory, which is a generalization of the classical differential subordination notion. This article employs a new integral operator introduced using [...] Read more.

The notion of the fuzzy set was incorporated into geometric function theory in recent years, leading to the emergence of fuzzy differential subordination theory, which is a generalization of the classical differential subordination notion. This article employs a new integral operator introduced using the class of meromorphic functions, the notion of convolution, and the Hurwitz–Lerch Zeta function for obtaining new fuzzy differential subordination results. Furthermore, the best fuzzy dominants are provided for each of the fuzzy differential subordinations investigated. The results presented enhance the approach to fuzzy differential subordination theory by giving new results involving operators in the study, for which starlikeness and convexity properties are revealed using the fuzzy differential subordination theory. Full article

(This article belongs to the Special Issue Advanced Research in Complex Analysis Operators and Special Classes of Analytic Functions)

12 pages, 274 KiB

Open AccessFeature PaperArticle

Series over Bessel Functions as Series in Terms of Riemann’s Zeta Function

by Slobodan B. Tričković and Miomir S. Stanković

Mathematics 2024, 12(19), 3000; https://doi.org/10.3390/math12193000 - 26 Sep 2024

Viewed by 664

Abstract

Relying on the Hurwitz formula, we find closed-form formulas for the series over sine and cosine functions through the Hurwitz zeta functions, and using them and another summation formula for trigonometric series, we obtain a finite sum for some series over the Riemann [...] Read more.

Relying on the Hurwitz formula, we find closed-form formulas for the series over sine and cosine functions through the Hurwitz zeta functions, and using them and another summation formula for trigonometric series, we obtain a finite sum for some series over the Riemann zeta functions. We apply these results to the series over Bessel functions, expressing them first as series over the Riemann zeta functions. Full article

15 pages, 295 KiB

Open AccessFeature PaperArticle

On Closed Forms of Some Trigonometric Series

by Slobodan B. Tričković and Miomir S. Stanković

Axioms 2024, 13(9), 631; https://doi.org/10.3390/axioms13090631 - 14 Sep 2024

Viewed by 765

Abstract

We have derived alternative closed-form formulas for the trigonometric series over sine or cosine functions when the immediate replacement of the parameter appearing in the denominator with a positive integer gives rise to a singularity. By applying the Choi–Srivastava theorem, we reduce these [...] Read more.

We have derived alternative closed-form formulas for the trigonometric series over sine or cosine functions when the immediate replacement of the parameter appearing in the denominator with a positive integer gives rise to a singularity. By applying the Choi–Srivastava theorem, we reduce these trigonometric series to expressions over Hurwitz’s zeta function derivative. Full article

(This article belongs to the Special Issue Special Functions and Related Topics)

23 pages, 539 KiB

Open AccessArticle

On Convoluted Forms of Multivariate Legendre-Hermite Polynomials with Algebraic Matrix Based Approach

by Mumtaz Riyasat, Amal S. Alali, Shahid Ahmad Wani and Subuhi Khan

Mathematics 2024, 12(17), 2662; https://doi.org/10.3390/math12172662 - 27 Aug 2024

Viewed by 896

Abstract

The main purpose of this article is to construct a new class of multivariate Legendre-Hermite-Apostol type Frobenius-Euler polynomials. A number of significant analytical characterizations of these polynomials using various generating function techniques are provided in a methodical manner. These enactments involve explicit relations [...] Read more.

The main purpose of this article is to construct a new class of multivariate Legendre-Hermite-Apostol type Frobenius-Euler polynomials. A number of significant analytical characterizations of these polynomials using various generating function techniques are provided in a methodical manner. These enactments involve explicit relations comprising Hurwitz-Lerch zeta functions and

λ

-Stirling numbers of the second kind, recurrence relations, and summation formulae. The symmetry identities for these polynomials are established by connecting generalized integer power sums, double power sums and Hurwitz-Lerch zeta functions. In the end, these polynomials are also characterized Svia an algebraic matrix based approach. Full article

(This article belongs to the Section E: Applied Mathematics)

► Show Figures

Figure 1

13 pages, 283 KiB

Open AccessArticle

The Mean Square of the Hurwitz Zeta-Function in Short Intervals

by Antanas Laurinčikas and Darius Šiaučiūnas

Axioms 2024, 13(8), 510; https://doi.org/10.3390/axioms13080510 - 28 Jul 2024

Cited by 4 | Viewed by 919

Abstract

The Hurwitz zeta-function

ζ (s, α)

,

s = σ + i t

, with parameter

0 < α ⩽ 1

is a generalization of the Riemann zeta-function

ζ (s)

( [...] Read more.

The Hurwitz zeta-function

ζ (s, α)

,

s = σ + i t

, with parameter

0 < α ⩽ 1

is a generalization of the Riemann zeta-function

ζ (s)

(

ζ (s, 1) = ζ (s)

) and was introduced at the end of the 19th century. The function

ζ (s, α)

plays an important role in investigations of the distribution of prime numbers in arithmetic progression and has applications in special function theory, algebraic number theory, dynamical system theory, other fields of mathematics, and even physics. The function

ζ (s, α)

is the main example of zeta-functions without Euler’s product (except for the cases

α = 1

,

α = 1 / 2

), and its value distribution is governed by arithmetical properties of

α

. For the majority of zeta-functions,

ζ (s, α)

for some

α

is universal, i.e., its shifts

ζ (s + i τ, α)

,

τ \in R

, approximate every analytic function defined in the strip

{s : 1 / 2 < σ < 1}

. For needs of effectivization of the universality property for

ζ (s, α)

, the interval for

τ

must be as short as possible, and this can be achieved by using the mean square estimate for

ζ (σ + i t, α)

in short intervals. In this paper, we obtain the bound

O (H)

for that mean square over the interval

[T - H, T + H]

, with

T^{27 / 82} ⩽ H ⩽ T^{σ}

and

1 / 2 < σ ⩽ 7 / 12

. This is the first result on the mean square for

ζ (s, α)

in short intervals. In forthcoming papers, this estimate will be applied for proof of universality for

ζ (s, α)

and other zeta-functions in short intervals. Full article

(This article belongs to the Special Issue The Pythagorean Heritage: From Number Theory and Combinatorics to Artificial Intelligence)

12 pages, 1495 KiB

Open AccessArticle

Geometric Features of the Hurwitz–Lerch Zeta Type Function Based on Differential Subordination Method

by Faten F. Abdulnabi, Hiba F. Al-Janaby, Firas Ghanim and Alina Alb Lupaș

Symmetry 2024, 16(7), 784; https://doi.org/10.3390/sym16070784 - 21 Jun 2024

Cited by 1 | Viewed by 1535

Abstract

The interest in special complex functions and their wide-ranging implementations in geometric function theory (GFT) has developed tremendously. Recently, subordination theory has been instrumentally employed for special functions to explore their geometric properties. In this effort, by using a convolutional structure, we combine [...] Read more.

The interest in special complex functions and their wide-ranging implementations in geometric function theory (GFT) has developed tremendously. Recently, subordination theory has been instrumentally employed for special functions to explore their geometric properties. In this effort, by using a convolutional structure, we combine the geometric series, logarithm, and Hurwitz–Lerch zeta functions to formulate a new special function, namely, the logarithm-Hurwitz–Lerch zeta function (LHL-Z function). This investigation then contributes to the study of the LHL-Z function in terms of the geometric theory of holomorphic functions, based on the differential subordination methodology, to discuss and determine the univalence and convexity conditions of the LHL-Z function. Moreover, there are other subordination and superordination connections that may be visually represented using geometric methods. Functions often exhibit symmetry when subjected to conformal mappings. The investigation of the symmetries of these mappings may provide a clearer understanding of how subordination and superordination with the Hurwitz–Lerch zeta function behave under different transformations. Full article

(This article belongs to the Special Issue Symmetries of Difference Equations, Special Functions and Orthogonal Polynomials II)

15 pages, 293 KiB

Open AccessArticle

A Joint Limit Theorem for Epstein and Hurwitz Zeta-Functions

by Hany Gerges, Antanas Laurinčikas and Renata Macaitienė

Mathematics 2024, 12(13), 1922; https://doi.org/10.3390/math12131922 - 21 Jun 2024

Cited by 2 | Viewed by 920

Abstract

In the paper, we prove a joint limit theorem in terms of the weak convergence of probability measures on

C^{2}

defined by means of the Epstein

ζ (s; Q)

and Hurwitz

ζ (s, α)

zeta-functions. The [...] Read more.

In the paper, we prove a joint limit theorem in terms of the weak convergence of probability measures on

C^{2}

defined by means of the Epstein

ζ (s; Q)

and Hurwitz

ζ (s, α)

zeta-functions. The limit measure in the theorem is explicitly given. For this, some restrictions on the matrix Q and the parameter

α

are required. The theorem obtained extends and generalizes the Bohr-Jessen results characterising the asymptotic behaviour of the Riemann zeta-function. Full article

13 pages, 297 KiB

Open AccessArticle

The Generalized Eta Transformation Formulas as the Hecke Modular Relation

by Nianliang Wang, Takako Kuzumaki and Shigeru Kanemitsu

Axioms 2024, 13(5), 304; https://doi.org/10.3390/axioms13050304 - 2 May 2024

Cited by 1 | Viewed by 1597

Abstract

The transformation formula under the action of a general linear fractional transformation for a generalized Dedekind eta function has been the subject of intensive study since the works of Rademacher, Dieter, Meyer, and Schoenberg et al. However, the (Hecke) modular relation structure was [...] Read more.

The transformation formula under the action of a general linear fractional transformation for a generalized Dedekind eta function has been the subject of intensive study since the works of Rademacher, Dieter, Meyer, and Schoenberg et al. However, the (Hecke) modular relation structure was not recognized until the work of Goldstein-de la Torre, where the modular relations mean equivalent assertions to the functional equation for the relevant zeta functions. The Hecke modular relation is a special case of this, with a single gamma factor and the corresponding modular form (or in the form of Lambert series). This has been the strongest motivation for research in the theory of modular forms since Hecke’s work in the 1930s. Our main aim is to restore the fundamental work of Rademacher (1932) by locating the functional equation hidden in the argument and to reveal the Hecke correspondence in all subsequent works (which depend on the method of Rademacher) as well as in the work of Rademacher. By our elucidation many of the subsequent works will be made clear and put in their proper positions. Full article

(This article belongs to the Section Algebra and Number Theory)

29 pages, 1484 KiB

Open AccessArticle

On the Sums over Inverse Powers of Zeros of the Hurwitz Zeta Function and Some Related Properties of These Zeros

by Sergey Sekatskii

Symmetry 2024, 16(3), 326; https://doi.org/10.3390/sym16030326 - 7 Mar 2024

Viewed by 1298

Abstract

Recently, we have applied the generalized Littlewood theorem concerning contour integrals of the logarithm of the analytical function to find the sums over inverse powers of zeros for the incomplete gamma and Riemann zeta functions, polygamma functions, and elliptical functions. Here, the same [...] Read more.

Recently, we have applied the generalized Littlewood theorem concerning contour integrals of the logarithm of the analytical function to find the sums over inverse powers of zeros for the incomplete gamma and Riemann zeta functions, polygamma functions, and elliptical functions. Here, the same theorem is applied to study such sums for the zeros of the Hurwitz zeta function

ζ (s, z)

, including the sum over the inverse first power of its appropriately defined non-trivial zeros. We also study some related properties of the Hurwitz zeta function zeros. In particular, we show that, for any natural N and small real

ε

, when z tends to n = 0, −1, −2… we can find at least N zeros of

ζ (s, z)

in the

ε

neighborhood of 0 for sufficiently small

| z + n |

, as well as one simple zero tending to 1, etc. Full article

(This article belongs to the Special Issue Functional Analysis, Fractional Operators and Symmetry/Asymmetry: Second Edition)

► Show Figures

Figure 1

19 pages, 324 KiB

Open AccessArticle

Binomial Series Involving Harmonic-like Numbers

by Chunli Li and Wenchang Chu

Axioms 2024, 13(3), 162; https://doi.org/10.3390/axioms13030162 - 29 Feb 2024

Cited by 3 | Viewed by 1453

Abstract

By computing definite integrals, we shall examine binomial series of convergence rate

\pm 1 / 2

and weighted by harmonic-like numbers. Several closed formulae in terms of the Riemann and Hurwitz zeta functions as well as logarithm and polylogarithm functions will be established, [...] Read more.

By computing definite integrals, we shall examine binomial series of convergence rate

\pm 1 / 2

and weighted by harmonic-like numbers. Several closed formulae in terms of the Riemann and Hurwitz zeta functions as well as logarithm and polylogarithm functions will be established, including a conjectured one made recently by Z.-W. Sun. Full article

(This article belongs to the Special Issue Research in Special Functions)

Search Results (58)

Further Information

Guidelines

MDPI Initiatives

Follow MDPI

Saved Queries

Search Filter Reset All

Years

Feature Papers

Subjects

Journals

Article Types

Countries / Regions

Search Results (58)

Further Information

Guidelines

MDPI Initiatives

Follow MDPI